The rate of microtubule breaking increases exponentially with curvature

Microtubules, cylindrical assemblies of tubulin proteins with a 25 nm diameter and micrometer lengths, are a central part of the cytoskeleton and also serve as building blocks for nanobiodevices. Microtubule breaking can result from the activity of severing enzymes and mechanical stress. Breaking can lead to a loss of structural integrity, or an increase in the numbers of microtubules. We observed breaking of taxol-stabilized microtubules in a gliding motility assay where microtubules are propelled by surface-adhered kinesin-1 motor proteins. We find that over 95% of all breaking events are associated with the strong bending following pinning events (where the leading tip of the microtubule becomes stuck). Furthermore, the breaking rate increased exponentially with increasing curvature. These observations are explained by a model accounting for the complex mechanochemistry of a microtubule. The presence of severing enzymes is not required to observe breaking at rates comparable to those measured previously in cells.


This supplementary information contains eight parts:
(1) Simulating microtubule trajectories and generating Figure 2 (2) Selection of pinned microtubules Microtubule trajectories were simulated according to the persistent random walk model. 1 Each point along the trajectory was described by three numbers: the x position, X, the y position, Y, and the direction of movement relative to the x-axis, . In a timestep Δt, the position of the tip of a microtubule gliding at velocity v with persistence length LP was updated according to: Where Norm(0,V) denotes a normal random variable with mean 0 and variance V. 100 microtubule trajectories of length 1 mm were generated according to the above equations using a discretization of Δ = Δ = 1 nm. To model the effect of discrete pixels, the generated sequence of (X,Y) values was then corrupted by noise which is uniformly distributed on ∈ [− , ]: , ~[− , ] The angle change distributions for different segment lengths used to calculate the interquartile range in Figure 2 of the main text were generated by the following procedure: First, we generated a set of 400 segment lengths, linearly spaced between 10 nm and 4 m. For each segment length, Δ , the corrupted MT trajectories defined by {( 1, , 1, ), … , ( , , , )} (9) (where we have abbreviated 'corrupted' to 'c' for brevity), are subsampled to generate decimated trajectories: where m is the number of points in the decimated trajectory and the indices or chosen iteratively such that: where denotes the point ( , , , ). In summary, the distance between consecutive points in the decimated trajectory is at least Δ .
The distribution of the change in angle between consecutive points along the decimated trajectory was estimated using the three-point method. In the three-point method, the change in angle, Δ , between points −1 , , and +1 is estimated by: The estimated curvature ( ) is given by: = Δ /Δ .
To obtain the 'Simulation' curve (dashed line) in Figure 2 of the main text, the sets of Δ | =2 −1 from each of the 100 corrupted MT trajectories were pooled together. The interquartile range (IQR) of the pooled set was recorded and associated with the value of Δ that was used to generate the decimated MT trajectory. The entire process (trajectory decimation, angle change calculation, pooling) was repeated for all 400 values of Δ and the resulting IQR estimates were plotted against the values of Δ in Figure 2 of the main text.
For the 'Original Data' curve in Figure 2 of the main text, the same procedure was repeated as for the simulation, only with simulated trajectories being replaced with experimental data. (

2) Selection of pinned microtubules
For the final breaking rate calculation, we manually selected all microtubules in the field-of-view that were pinned to inactive kinesin. A pinning event is identified if the tip of the microtubule is stationary in multiple frames while the body of the microtubules moves. We notice two types of pinning events: spiraling microtubules where the tip is able to rotate and fishtailing microtubules that are unable to rotate. However, there are some microtubules that become pinned and break free again within the time between frames. When this occurs, we can see the microtubule suddenly bend away from its original trajectory. The sharp bend due to pinning is used to identify these brief (

3) Maximum likelihood estimation of breaking rate
The parameters for the breaking rate equation were fit using maximum likelihood estimation of the breaking probabilities of the observed microtubules. The breaking probability as a function of curvature can be written as: Where: The likelihood is the product of the probabilities of all breaking and non-breaking events: By taking the natural log of Equation 3 and inserting Equations 1-2, the log-likelihood is obtained: Equation 4 was maximized to find the best fit for parameters a and b. Asymmetric confidence intervals (95%) were constructed by evaluating likelihood profiles for each parameter. The fitted parameters for each individual experiment and for the pooled data are listed in Supplementary

(4) Length dependence of breaking rate
To test if the parameters depend on the microtubule length, we split the population of microtubules into two equally large groups based on their length with one population containing all microtubule segments from microtubules shorter than 6 μm and the other containing all microtubule segment from microtubules longer than 6 μm. The same maximum likelihood estimation procedure described in Supplementary

(5) Time dependence of breaking rate
To test if the parameters depend on time of the breaking event, we split the population of microtubules into two equally large groups with one population containing all microtubule segments from microtubules in the first half of each recording (t < 900 s) and the other containing all microtubule segment from microtubules in the second half of each recording (t ≥ 900 s). The same maximum likelihood estimation procedure described in Supplementary Fig. 3a). For the next five to ten micrometers from the attachment points, the kinesins are pulling perpendicularly to the axis as the buckling microtubule moves sideways. For the rupture force under perpendicular loading, Khataee and Howard 12 give a force-dependent rate equation and associated parameters implying that the force required to unbind kinesin from the microtubule in 30 ms (roughly the time required to stretch the kinesin tail to full length when the microtubule is moving at our velocity of 900 nm/s) is 30 pN. So the forces exerted by the kinesins on their tubulin attachment points is expected to increase (or at least not decrease) with the distance from the attachment point as the force changes from axial loading with the stall force to perpendicular loading with the rupture force. However, the breaking probability rapidly decreases with the distance from the attachment point, which is inconsistent with the force profile along the microtubule.
Triclin et al. 13 propose that already the internal forces exerted by kinesin walking along microtubules can enhance the removal of tubulins from the lattice five-fold. Kuo et al. 14 similarly measured that a single kinesin pulling with its stall force (4-7 pN) accelerates tubulin removal about five-fold. Thus, kinesin forces enhance the probability of tubulin removal, but are not the main factor responsible for the distribution of breaking sites and the acceleration of breaking.
Aging: Another potential explanation for the difference between smoothly gliding and pinned microtubules is that the segments of a smoothly gliding microtubule quickly pass through a trajectory region with high curvature, while the segments of a pinned microtubule remain highly curved for an extended time. If breaking is facilitated by a recent history of high curvature (causing "aging"), this could increase the breaking probability for pinned microtubules. We tested this by reviewing the recent history of breaking events. For each breaking event, we compared the curvature during the breaking event to the curvature of the same microtubule segment in the preceding frame (Supplementary Figure 4a)

(8) Equilibrium Curvature Distribution Due to Breaking
The aim of this calculation is to determine the steady state curvature distribution of a population of microtubules which is subject to a curvature-reducing process and a curvature-producing process. The curvature-reducing process is breaking, which removes segments with a given curvature with a rate that exponentially increases with the segments curvature. The curvatureproducing process is either the extension of the segment length resulting from motor propulsion  If we consider a microtubule segment as a circular arc segment between two attachment points, the distance between the two points, a, can be described by the radius R and the arc angle α: (1) The length of the microtubule segment is equivalent to the arc length of the osculating circle: Increasing the arc length by Δs while maintaining the distance between the attachment points leads to a relationship between the radius of curvature before (R) and after (R-ΔR) the increase in the arc length: If the radius of curvature is expressed as the inverse of the curvature κ, eq. (3) After neglecting higher order terms and using Δa ≈ Δs, equation (6)  using the 95% confidence interval of the characteristic breaking radius, r*.

Supplementary Video 1-4 | Gliding Motility Assays.
Videos of the gliding assays of fluorescently labeled microtubules gliding on surfaces coated with kinesin-1. Each video number corresponds to the experiment number referenced in the main text (Figures 3 and 4). Video speed: 50x accelerated relative to real time. Field of View: 133.12 μm x 133.12 μm.

Supplementary Data 1 | Pinned Microtubule Location.
Excel file containing data on each manually selected nonbreaking pinned microtubule. Included is the location of the pinned tip and the frame the microtubules become pinned and unpinned.
Data are separated into sheets corresponding to the experiment number referenced in the main text (Figures 3 and 4).

Supplementary Data 2 | Breaking Microtubule Location
Excel file containing data on each manually selected breaking microtubule. Included is the breaking point of the microtubule, the frame in which the break occurs, whether the break was the result of a pinning event and how long the microtubule was pinned before the breaking event.
Data are separated into sheets corresponding to the experiment number referenced in the main text (Figures 3 and 4).